All posts by Asaf Karagila

Back in the days of yore, if one wanted to know mathematics, one would have to go to the university and take a course; or hire a tutor; or go to the library and open a book and learn on their own.

And that was fine. All three options are roughly equivalent, in the sense that they present you the material in a very structured way (or they at least intend to). You don’t reach the definition of $\aleph_0$ because you defined what is equipotency and cardinality. You don’t reach the definition of a derivative before you have some semblance of notion of continuity. Knowledge was built in a very structural way. Sometimes you use crutches (e.g. some naive understanding of the natural numbers before you formally introduce them later on as finite ordinals), but for the most part there is a method to the madness.

Fast forward to the information age. Everything is one Wikipedia, every entry tries to be self-contained with respect to at least a short introduction. You can now learn about Hilbert’s Grand Hotel (and his shrewd business acumen), without learning what it means for two sets to have the same cardinality. And that is an essential gap. Yes, the point of the Grand Hotel is to demonstrate that infinite sets can have different properties than finite sets when it comes to cardinality. And yes, depending on the teacher, this can be a segue into the definition of cardinality (although in my opinion not as good as the usual “do I have the same amount of fingers on each hand without counting them?” approach). But nevertheless, in an unstructured learning environment there is a high risk—which is actual reality, as witnessed by the many confused questions on the internet regarding infinity and the Grand Hotel—that the reader is not going to follow through with the definition of cardinality, since this example will already be confusing enough, or distracting enough from being just an example.

Another terrible example is the old Numberphile video about $1+2+3+\ldots=-\frac1{12}$. Yes, this can be found in many books and so on. But in all these books, I am sure, it will be mentioned explicitly that this manipulation is not the standard definition of summation, but rather obtained through other mathematically valid methods that have been subjected to abuse of notation. Stripping the context from all this, and just presenting this summation as a magic trick, is a surefire way to confuse everyone who is not already familiar enough with these topics. And of course that it has, I even had students of mine asking me about that back when the video first hit tsunami sized waves across the web.

What’s the problem, you might ask? Let those people go online and ask experts! Well, it turns out that there is a reason you don’t talk about Ramanujan summation or zeta regularization in the first semester of undergrad. And people come with an honest question, and they expect an easy answer to quickly dispel the dissonance they have between this weird summation and what they know (or think they know). And there are no quick answers which are clear, simple, and not entirely condescending. There is a reason why one has to work through several years of set theory before gaining the actual and intuitive understanding why you need the axiom of choice to prove there is an injection from $\omega_1$ into the real numbers. These things are complicated.

Dangerous knowledge usually refers to knowledge that is considered dangerous for other people to have. Like how at some point terrorist organizations realized that if you just teach everyone to make homemade bombs, it’s going to be a lot harder to actually stop the bomb production and hinder the organization (and even caused people who just self-identified with the cause of the organization to pick up arms and commit terrible acts).

But in the context of education, I think that a dangerous knowledge is knowledge which you obtain without a structured set up. You are not ready for that sort of knowledge, and you do not have the means of placing it in the bigger picture. I had this problem, through all my life, I have gone to read about things, and I skipped and jumped ahead, and I tried to learn further and better. And every time I jumped and made an unstructured “discovery” I eventually had to go back and correct the err of my ways.

The question, from an educational point of view, is how can you fight this? How can you make sure that dangerous knowledge is kept to a minimum?

One way is to instil into students from a very young age the sense of curiosity and wonderment. I remember reading somewhere about someone who as a kid opened up a book, and read about some problem, then started to work backwards to obtain all the knowledge necessary for understanding it better. It could have been Feynman or Wiles, I am not sure, and it doesn’t matter. The point is that when coming across dangerous knowledge, the protagonist of that story “defused” the danger by starting to go backwards and learning the necessary framework.

In today’s modern era, where everything needs to be a click-bait-bite-size-immediately-satisfying thing, the above is difficult. It is hard to make sure that people actually sit down to read. People want the information they feel is missing, and not a long list of information they are actually missing. And not to mention that re-educating the whole planet seems like a fairly Herculean task.

But I do think that at least in academia this is possible. It should be possible to try and educate students about this. I think it is important, especially in natural sciences, where there are good chances that the students will go on to research later (either in academia, or elsewhere) to remember this. Because having dangerous knowledge can affect the way you perceive your actual knowledge. It can re-frame your knowledge incorrectly, or shift the importance of something you are currently learning (or about to) from one side of the picture to another, and not necessarily in a good way.

Another option is to educate people about the existence and dangers of dangerous knowledge. Once you are aware that learning something in an unstructured way can be problematic, you can put this knowledge in check automatically, reminding to yourself that you need to know more in order to fully appreciate some anecdotal piece of information that you read online, and heard about. This can also motivate you to go and actually study more about something, which is always a good outcome.

There are many discussions on the multiverse of set theory generated by a model. The generic multiverse is given by taking all the generic extensions and grounds of some countable transitive model.

Hamkins’ multiverse is essentially taking a very ill-founded model and closing it to forcing extensions, thus obtaining a multiverse which is more of a philosophical justification, for example every model is a countable model in another one, and every model is ill-founded by the view of another model. The problem with this multiverse is that if we remove the requirement for genericity, then everything else can be satisfied by the same model. Namely, $\{(M,E)\}$ would be an entire multiverse. That’s quite silly. Moreover, we sort of give up on a concrete notion of natural numbers that way, and this seems a bit… off putting.

There is also Väänänen’s multiverse, which is more abstractly defined, and I cannot for the life of me recall its definition and its details.

Some time ago Ur Ya’ar gave a seminar talk about Hamkins’ multiverse in the logic seminar in Jerusalem. It was interesting, and afterwards Yair Hayut and myself talked with Ur about these multiverses. One idea that came up, and I don’t think that I ever ran into it, is sort of a combination between the generic multiverse and Hamkins’ multiverse. Consider the following axiom “Every real is an element of a transitive model”. Now look at $\cal M$, the set of all the countable transitive models, we get the following axioms are satisfied by $\cal M$:

If $M\in\cal M$, then every generic extension and every ground of $M$ is also in $\cal M$.

If $M\in\cal M$, then every inner model of $M$ is also in $\cal M$.

If $M\in\cal M$, then there is some $N\in\cal M$ such that $M\in N$ and $N\models M\text{ is countable}$.

For all $M,N\in\cal M$, $L^M$ and $L^N$ are comparable.

So what do we have here? We have a multiverse of sets, it is closed under generic extensions and grounds, and it is even closed under definable inner models. It also has the property that we can always find bigger models that think a given model is countable.

Now, I have no idea what useful things can come out of this multiverse. And I would imagine that one should first refine this notion a bit more before it becomes actually useful for something. But nonetheless, it seems like an interesting interpretation of the whole notion of multiverse.

I am sitting in the 6th European Set Theory Conference in Budapest, and watching all these wonderful talks, and many of them use colors for emphasis of some things. But yesterday one of the talks was using “too many colors”, enough to make me make a comment at the end of the talk after all the questions were answered. Since I received some positive feedback from other people here, I decided to write about it on my blog, if only to raise some awareness of the topic.

There is a nontrivial percentage of the population which have some sort of color vision deficiency. Myself included. Statistically, I believe, if you have 20 male participants, then one of them is likely to have some sort of color vision issues. Add this to the fairly imperfect color fidelity of most projectors, and you get something that can be problematic.

Now, I’m not saying “don’t use any colors”. Not at all. Just keep in mind that some people might have problems with your choice of colors. Using too many colors can be distracting, and one of the slides in the said talk had black text almost on par with the rest of the colored text. This is far from ideal. But since color deficiency can vary from one to another, let me only give an account of my own personal experience. I cannot do anything more, after all.

I have a mild red-green issue. But this means also that yellow and bright green, or light orange, all mix together sometimes; and darker greens can be red or brown (which themselves are often mixed); and blues can mix with purple, and sometimes with pink as well. One other effect of color deficiency is that you are more sensitive to brightness and darkness (the eye compensates the damaged cones by having better rods, so your night vision gets somewhat better, for example).

So when you have a slide with some pink/purple and green/yellow/orange and some blue and some red and some black, my brain will not read the text. My brain will try to make sense of the colors. Not to mention the terrible eye strain coming from the brighter colors (here the quality of the viewing media is important, I’m sure that I’d be fine watching the same slides on a proper computer monitor). There were slides that I had to turn my eyes away from the talk. Yes, it was pretty bad.

What can you do about it? Don’t use colors when you don’t have to. Use boldface or italics for emphasis when possible, or different font family entirely. If you want to use colors, using them sparingly, and try to avoid relatively close colors together and certainly try to avoid brighter colors like light green or yellow. If you know a color blinded person, you can maybe ask them to give some critique on your choice of colors.

Some people commented to me after my remark that they prefer the colors, and they are helpful. I understand that. Again, the point is not to get people to use colors. Just… to use them intelligently. Colors are like spices. I’m not trying to get you to cook without spices, but you’re not going to serve a dish entirely made of cinnamon and cumin.

In the name of all color vision deficient people, thanks in advance for your consideration!

When one is ascending a difficult path uphill, it is a good idea to keep your eyes on the path as you move forward. However, it is not a bad idea to stop sometimes, look back, and appreciate the beauty of the ground you have already covered.

This understanding extends to any difficult task which is completed in a long series of steps.

As some of you may have noticed, I don’t use this blog to write about my papers in the “traditional way” math bloggers summarize and explain their recent work. I think my papers are prosaic enough to do that on their own. I do use this blog as an outlet when I have to complain about the arduous toil of being a mathematician (which has an immensely bright light side, of course, so in the big picture I’m quite happy with it).

This morning I woke up to see that my paper about the Bristol model was announced on arXiv. But unbeknownst to the common arXiv follower, this also marks the end of my thesis. The Hebrew University is kind enough to allow you to just stitch a bunch of your papers (along with an added introduction) and call it a thesis. And by “stitch” I mean literally. If they were published, you’re even allowed to use the published .pdf (on the condition that no copyright infringement occurs).

My dissertation is composed of three papers, all of which are on arXiv (links in the “Papers” page of this site):

Of course, the ideal situation is that all three papers have been accepted for publication, but all three of them are still under review. So it puts me at this odd situation where I will have essentially four sets of referees (one for each paper, and then two additional referees for my thesis), and so the output can end up oddly different between the resulting dissertation and the published papers. But that’s fine.

In any case. Those of you who are interested in reading my thesis can find it in those three papers. I am probably going to post the final thesis online when it will be approved, but the only thing you’re currently missing out is an introduction with some minor historical background and a summary of the three papers. So if you read all three, you don’t really need that introduction anyway.

Good. So what next? I have a few things lined up. More news will follow as reality unfolds itself like a reverse origami.

This is not something particularly interesting, I think. But it’s a nice exercise in Fodor’s lemma.

Theorem. Suppose that $\kappa$ is regular and uncountable, and $\pi\colon\kappa\to\kappa$ is a bijection mapping stationary sets to stationary sets. Then there is a club $C\subseteq\kappa$ such that $\pi\restriction C=\operatorname{id}$.

Proof. Note that the set $\{\alpha\mid\pi(\alpha)

This is just something I was thinking about intermittently for the past few years, but now I finally spent enough energy to figure it out. And it’s cute. (Soon I will post more substantial posts, on far more exciting topics! Don’t worry!)

Good news! I’m about to finish my dissertation. Hopefully, come summer I will be Dr. Asaf Karagila.

So the next order of business is finding a position for next year. So far nothing came up. But I’m open to hearing from the few readers of my blog if they know about something, or have some offers that might be suitable for me.

My mentor, teacher, mathematical confidant and generally good friend, Matti Rubin passed away this morning. Many of the readers here know him for his mathematical work, many knew him as a friend as well, or as a teacher.

Matti was a kind teacher, even if sometimes over-pedantic.

Recently he was diagnosed with lung cancer. I spoke with him two weeks ago, shortly before I went to the Arctic Set Theory conference, and he sounded positive. He felt that he might not survive this battle, but he was optimistic about the short time era. I was hoping to talk to him once more and to tell him about a student of mine that he would have surely liked.

Alas, this morning Matti left this world. Apparently there had been a sudden deterioration of his condition, and this morning, tired and hurt he left us here. I take solace in the fact that I have known him. I had learned from him. I have created a minuscule copy of the original in my mind, to remind me where I’m being mathematically dishonest while writing proofs (“This is not obvious, and that is not clear”).

He will be sorely missed by all of us here.

A photo I took from his logic course back in the Spring of 2009. Probably talking about model theory or Turing machines…

If you follow my blog, you probably know that I am a big fan of Michael Stevens from the VSauce channel, who in the recent year or so released several very good videos about mathematics, and about infinity in particular. Not being a trained mathematician, Michael is doing an incredible task.

Non-mathematicians often tend to be Platonists “by default”, so they will assume that every question has an answer and sometimes it’s just that we don’t know that answer. But it’s out there. It’s a fine approach, but it can somewhat fly in the face of independence if you are not trained to think about the difference between true and provable.

This morning, as I was watching the new video of Physics Girl, there was an announcement about a new math channel. So of course I went to look into that channel. It’s fairly new, and there are only a handful of videos, but they already tackled some nice topics. The videos are written and hosted by a Cornell grad student, Kelsey Houston-Edwards. I watched the one about a hierarchy of infinities, and while I was a bit skeptic after the first minute, I was quite happy at the end, when the discussion went from just the fact that the reals are uncountable (although without a proof, and that’s fine, I guess, there are plenty of those on the internet), to a discussion about the continuum hypothesis.

In another video, Kelsey tackles mathematical Platonism and its somewhat-opposite, Formalism. And it’s done well. Kelsey doesn’t lean into one side or another, because at the end of the day, mathematicians—as opposed to mathematical philosophers—do mathematics, and that is their main concern. The philosophy is mostly a spice to add some taste and meaning to your work.

In any case, I enjoyed watching the few videos that I have, and I hope that you will as well. I’m sure that this is not the last that you’ll see me talk about Kelsey and her channel.

I’ve been given the chance to teach the course in axiomatic set theory in Jerusalem this semester. Today I gave my first lecture as a teacher. It went fine, I even covered more than I expected to, which is good, I guess. I am also preparing lecture notes, which I will probably post here when the semester ends. These predicated on some rudimentary understanding in logic and basic set theory, so there might be holes there to people unfamiliar with the basic course (at least the one that I gave with Azriel Levy for the past three years).

Yesterday, however, I spent most of my day thinking about how we—as a collective of set theorists—teach axiomatic set theory. About that usual course: axioms, ordinals, induction, well-founded sets, reflection, $V=L$ and the consistency of $\GCH$ and $\AC$, some basic combinatorics (clubs, Fodor’s lemma, maybe Solovay or even Silver’s theorem). Up to some rudimentary permutation.

Is this the right way to approach axiomatic set theory? This path is not easy to justify. Sure, you can justify things like well-founded sets by arguing that this is how we justify the Axiom of Foundation. And you could argue that this is a rudimentary foray into inner model theory, and that this is important. And you are absolutely right. But on the other hand, I feel that engaging the students should involve more set theory which is “interactive”. Where you obtain actual results, rather than just consistency of axioms, especially axioms which you have very little motivation towards.

I mean, look at how we teach (or learn) about algebraic structures. We don’t spend all semester just with the axioms of groups, or rings, proving things. We also see a lot of examples, and a lot of ways where these structures interact with mathematics. Set theory doesn’t have this luxury, we don’t have natural models to work with and their interactions with mathematics is meta-theoretical, rather than direct as it is the case with groups and rings.

So set theory, in essence, should be taught in a mixture of motivating examples and consistency proofs. I am taking this from my advisor, who is a wonderful teacher, as anyone who ever sat in his lectures could witness. A couple of years ago, Menachem gave a course about stationary tower forcing. In most texts about stationary tower forcing, you spend the first several dozen pages in technical concepts like Completely Jonsson cardinals, and so on. But Menachem started with the motivation: universally Baire sets, and their properties. Once you understand those, stationary tower forcing becomes much easier to digest, because it is with purpose. Last year, and next semester, Menachem is talking about inner models, and again a lot of motivation is given into fine structural considerations, mainly square’ish ones for the basic fine structure of $L$, but also through mice we get a good intuition as to what $K^{DJ}$ is supposed to be.

Right. So the basic axiomatic set theory course. What can we do about that? Well, my initial approach is to take $\ZF$ for starters. Motivate Foundation by talking about induction, and then prove that Foundation adds no new contradictions. After that, we’ll see exactly, but the next step is again motivation for either choice or Reflection principles. In either case, I feel that having motivation interspersed with consistency proofs is key here.

So now, let me ask you, my fellow set theorists, who have taught courses in axiomatic set theory. What is your experience on the matter? What is your take on my approach? This is my first time doing this, and I will definitely be reporting again during the semester and afterwards. But I also want to hear what you have to say on the matter. I will leave the comments open, but also feel free to contact me over email.